In a world where we get fleeced on a daily basis by large corporations, a world where people are duped into believing that "brands" equal status, I resolutely refuse to wear logos, unless I'm a) doing charity work, which happens frequently, or b) being paid for it.
The business model of "win something if you wear it" makes no sense. With cars, many competitions have been run along the lines of "put this bumper sticker on your car, and IF you get spotted, we'll read out your number plate on the radio, after which you have to phone us to claim your prize".
Bicycles, to point out the obvious, don't have number plates, so even if you run the contest on a "if you get spotted wearing this", how would you resolve that glimpse of a cyclist to an identifiable person? (If you DO have a workable method to do so, I suggest you contact the Daily Fail, who is forever ranting about how we need number plates!)
Even if you can get a reliable method to solve this conundrum, without getting strung up by cyclists for causing them to lose their anonymity, such competitions typically favour those in large, densely populated areas, such as London, Manchester or Birmingham, at the exclusion of smaller cities and towns, with those in completely rural areas standing no chance at all.
In view of the above, no, I don't think I'd be interested in your competition idea. I also don't believe most others will be interested, outside of the getting-a-free-cycling-top factor.